3.266 \(\int \frac{\log (c (d+e \sqrt{x})^p)}{f+g x^2} \, dx\)

Optimal. Leaf size=541 \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt{-\sqrt{-f}}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt{-\sqrt{-f}}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

[Out]

-(Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sq
rt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))]
)/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-
f]] - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e*
(-f)^(1/4) - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]]
- d*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))])
/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sqrt[-f]*
Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])

________________________________________________________________________________________

Rubi [A]  time = 0.813215, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2472, 275, 205, 2416, 260, 2394, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt{-\sqrt{-f}}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt{-\sqrt{-f}}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]

[Out]

-(Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sq
rt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))]
)/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-
f]] - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e*
(-f)^(1/4) - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]]
- d*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))])
/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sqrt[-f]*
Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])

Rule 2472

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q
, x], x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right )}{f+g x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \log \left (c (d+e x)^p\right )}{f+g x^4} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{\sqrt{g} x \log \left (c (d+e x)^p\right )}{2 \sqrt{-f} \left (\sqrt{-f} \sqrt{g}-g x^2\right )}-\frac{\sqrt{g} x \log \left (c (d+e x)^p\right )}{2 \sqrt{-f} \left (\sqrt{-f} \sqrt{g}+g x^2\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{g} \operatorname{Subst}\left (\int \frac{x \log \left (c (d+e x)^p\right )}{\sqrt{-f} \sqrt{g}-g x^2} \, dx,x,\sqrt{x}\right )}{\sqrt{-f}}-\frac{\sqrt{g} \operatorname{Subst}\left (\int \frac{x \log \left (c (d+e x)^p\right )}{\sqrt{-f} \sqrt{g}+g x^2} \, dx,x,\sqrt{x}\right )}{\sqrt{-f}}\\ &=-\frac{\sqrt{g} \operatorname{Subst}\left (\int \left (-\frac{\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} x\right )}+\frac{\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{\sqrt{-f}}-\frac{\sqrt{g} \operatorname{Subst}\left (\int \left (\frac{\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}-\frac{\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{\sqrt{-f}}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt{-\sqrt{-f}}-\sqrt [4]{g} x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt [4]{g}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}-\sqrt [4]{g} x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt [4]{g}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt{-\sqrt{-f}}+\sqrt [4]{g} x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt [4]{g}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}+\sqrt [4]{g} x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt [4]{g}}\\ &=-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} x\right )}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} x\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}\\ &=-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{g} x}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{g} x}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{g} x}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{g} x}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{2 \sqrt{-f} \sqrt{g}}\\ &=-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt{-\sqrt{-f}}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{Li}_2\left (-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{Li}_2\left (-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{Li}_2\left (\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt{-\sqrt{-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{Li}_2\left (\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}\\ \end{align*}

Mathematica [C]  time = 0.277251, size = 422, normalized size = 0.78 \[ \frac{p \text{PolyLog}\left (2,-\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \text{PolyLog}\left (2,\frac{i \sqrt [4]{g} \left (d+e \sqrt{x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+i e \sqrt [4]{-f}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \left (d+e \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )+\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )-\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt{x}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt{x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]

[Out]

(Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))] - Log[c*(d + e*Sq
rt[x])^p]*Log[(e*((-f)^(1/4) - I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Lo
g[(e*((-f)^(1/4) + I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - I*d*g^(1/4))] + Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(
1/4) + g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4))] + p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4)
- d*g^(1/4)))] - p*PolyLog[2, (I*g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - p*PolyLog[2, (g^(1/4
)*(d + e*Sqrt[x]))/(I*e*(-f)^(1/4) + d*g^(1/4))] + p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^
(1/4))])/(2*Sqrt[-f]*Sqrt[g])

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Maple [F]  time = 0.727, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{g{x}^{2}+f}\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)

[Out]

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e \sqrt{x} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(log((e*sqrt(x) + d)^p*c)/(g*x^2 + f), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e*x**(1/2))**p)/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e \sqrt{x} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(log((e*sqrt(x) + d)^p*c)/(g*x^2 + f), x)